[[159,37,5]] · QEC Challenge

[[159,37,5]] d ≤

159

kd²/n

5.81761

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 5, d_Z ≤ 5 · w_X = 6, w_Z = 6 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 5 · witness weight 5 (claimed upper_bound)

witness found by @mathys · research/kit/surrogate.lightest_logical · found at 1000 trials · survived 1000 trials · 2026-11-02

witness operator (support, 5 qubits)

[5, 14, 26, 45, 56]

d_Z 5 · witness weight 5 (claimed upper_bound)

witness found by @mathys · research/kit/surrogate.lightest_logical · found at 1000 trials · survived 1000 trials · 2026-11-02

witness operator (support, 5 qubits)

[3, 65, 66, 74, 86]

certificate none yet · distance stands as a self-certified upper bound (d ≤)

Diagnostics

computed by the verifier from the parity checks, the layout, and the stored witnesses; shown as evidence, not used for ranking

girth H_X 8 · H_Z 8 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 6 · H_Z 6

qubit degrees H_X 1–4 (mean 2.302) · H_Z 1–4 (mean 2.302)

trapping sets H_X (1,1)×10 (2,1)×30 (3,1)×90 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_X

(1,1): 10 (1,2): 118 (1,3): 4 (1,4): 27 (2,1): 30 (2,2): 347 (2,3): 62 (2,4): 417 (2,5): 12 (2,6): 47 (3,1): 90 (3,2): 1037 (3,3): 583 (3,4): 3876 (3,5): 463 (3,6): 2072 (3,7): 96 (3,8): 303

trapping sets H_Z (1,1)×10 (2,1)×30 (3,1)×90 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_Z

(1,1): 10 (1,2): 118 (1,3): 4 (1,4): 27 (2,1): 30 (2,2): 347 (2,3): 62 (2,4): 417 (2,5): 12 (2,6): 47 (3,1): 90 (3,2): 1037 (3,3): 583 (3,4): 3882 (3,5): 451 (3,6): 2078 (3,7): 96 (3,8): 303

Construction & provenance

authors @mathysrennela

provenance submitted through the challenge

novelty novelty not audited

construction priced calculus E-PUNCTURE transform of codes/160-36-6.json

model Beam 1.0 (claimed, not verified)

date 2026-11-02

family other (a tag, not a ranking)

locality unrestricted (computed from the layout)

weight class weight ≤ 6 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[159,37,5]] — priced calculus E-PUNCTURE transform of codes/160-36-6.json

Direction & hypothesis

Weight-6 unrestricted CSS cell. Transform a connected CSS parent (codes/160-36-6.json, a board entry) with the priced construction calculus to obtain a connected child that is not dominated by any existing entry in its track cell.

What was searched

Gap-targeted search: all priced calculus moves (peel, puncture, deletion, rebase) applied to board codes. Targets Pareto frontier gaps. Rule used: E-PUNCTURE. Parent: codes/160-36-6.json. Connected children only; bare-qubit children and dominated candidates dropped. Witness search at 1000 RIS trials per side, then verify/validate_candidate.py with full refutation.

Evidence trail

The submitted distance is a witness-backed upper bound: d = min(d_X, d_Z) with explicit X and Z logical witnesses carried in the JSON. Parent entry: codes/160-36-6.json.

Tools

Model: Beam 1.0. Harness: research/kit (coordination, submit, prescreen), code-calculus consumer.

Reproduction

Rebuild H_X/H_Z from codes/160-36-6.json by applying the E-PUNCTURE move of the priced calculus; re-run the witness search at the stated budget.

Parity checks

X-checks 61 (max weight 6) · Z-checks 61 (max weight 6)

H_X (61 checks, sparse supports)

[0, 29, 63, 75, 127, 142] [1, 30, 64, 76, 128, 155] [3, 28, 66, 78, 130, 157] [4, 17, 67, 79, 131, 146] [5, 18, 68, 80, 127, 132] [6, 19, 69, 81, 133, 144] [7, 16, 70, 82, 129, 134] [5, 8, 71, 83, 135, 150] [6, 9, 72, 84, 131, 136] [7, 10, 73, 85, 137, 148] [4, 11, 74, 86, 133, 138] [12, 25, 75, 87, 139, 154] [13, 26, 76, 88, 135, 140] [14, 27, 77, 89, 141, 152] [15, 24, 78, 90, 137, 142] [13, 16, 79, 91, 143, 158] [14, 17, 80, 92, 139, 144] [15, 18, 81, 93, 145, 156] [12, 19, 82, 94, 141, 146] [1, 20, 63, 83, 130, 147] [2, 21, 64, 84, 143, 148] [3, 22, 65, 85, 128, 149] [0, 23, 66, 86, 145, 150] [21, 24, 67, 87, 134, 151] [22, 25, 68, 88, 147, 152] [23, 26, 69, 89, 132, 153] [20, 27, 70, 90, 149, 154] [9, 28, 71, 91, 138, 155] [10, 29, 72, 92, 151, 156] [11, 30, 73, 93, 136, 157] [31, 60, 95, 107, 127, 149] [32, 61, 96, 108, 128, 134] [33, 62, 97, 109, 129, 147] [34, 59, 98, 110, 130, 132] [35, 48, 99, 111, 131, 153] [36, 49, 100, 112, 132, 138] [37, 50, 101, 113, 133, 151] [38, 47, 102, 114, 134, 136] [36, 39, 103, 115, 135, 157] [37, 40, 104, 116, 136, 142] [38, 41, 105, 117, 137, 155] [35, 42, 106, 118, 138, 140] [43, 56, 107, 119, 129, 139] [44, 57, 108, 120, 140, 146] [45, 58, 109, 121, 127, 141] [46, 55, 110, 122, 142, 144] [44, 47, 111, 123, 133, 143] [45, 48, 112, 124, 144, 150] [46, 49, 113, 125, 131, 145] [43, 50, 114, 126, 146, 148] [32, 51, 95, 115, 137, 147] [33, 52, 96, 116, 148, 154] [34, 53, 97, 117, 135, 149] [31, 54, 98, 118, 150, 152] [52, 55, 99, 119, 141, 151] [53, 56, 100, 120, 152, 158] [54, 57, 101, 121, 139, 153] [51, 58, 102, 122, 154, 156] [40, 59, 103, 123, 145, 155] [41, 60, 104, 124, 130, 156] [42, 61, 105, 125, 143, 157]

H_Z (61 checks, sparse supports)

[0, 5, 31, 45, 127, 150] [1, 22, 32, 62, 128, 147] [2, 7, 33, 43, 129, 148] [3, 20, 34, 60, 130, 149] [4, 9, 35, 49, 131, 138] [5, 26, 34, 36, 132, 135] [6, 11, 37, 47, 133, 136] [7, 24, 32, 38, 134, 137] [8, 13, 39, 53, 135, 158] [9, 30, 38, 40, 136, 155] [10, 15, 41, 51, 137, 156] [11, 28, 36, 42, 138, 157] [12, 17, 43, 57, 139, 146] [2, 13, 42, 44, 140, 143] [14, 19, 45, 55, 141, 144] [0, 15, 40, 46, 142, 145] [16, 21, 47, 61, 134, 143] [6, 17, 46, 48, 131, 144] [18, 23, 49, 59, 132, 145] [4, 19, 44, 50, 133, 146] [20, 25, 33, 51, 147, 154] [10, 21, 50, 52, 148, 151] [22, 27, 31, 53, 149, 152] [8, 23, 48, 54, 150, 153] [24, 29, 37, 55, 142, 151] [14, 25, 54, 56, 139, 152] [12, 27, 52, 58, 141, 154] [1, 28, 41, 59, 130, 155] [18, 29, 58, 60, 127, 156] [3, 30, 39, 61, 128, 157] [63, 68, 95, 109, 127, 147] [64, 85, 96, 126, 128, 148] [65, 70, 97, 107, 129, 149] [66, 83, 98, 124, 130, 150] [67, 72, 99, 113, 131, 151] [68, 89, 98, 100, 132, 152] [69, 74, 101, 111, 133, 153] [70, 87, 96, 102, 134, 154] [71, 76, 103, 117, 135, 155] [72, 93, 102, 104, 136, 156] [73, 78, 105, 115, 137, 157] [74, 91, 100, 106, 138, 158] [75, 80, 107, 121, 127, 139] [65, 76, 106, 108, 128, 140] [77, 82, 109, 119, 129, 141] [63, 78, 104, 110, 130, 142] [79, 84, 111, 125, 131, 143] [69, 80, 110, 112, 132, 144] [81, 86, 113, 123, 133, 145] [67, 82, 108, 114, 134, 146] [83, 88, 97, 115, 135, 147] [73, 84, 114, 116, 136, 148] [85, 90, 95, 117, 137, 149] [71, 86, 112, 118, 138, 150] [87, 92, 101, 119, 139, 151] [77, 88, 118, 120, 140, 152] [89, 94, 99, 121, 141, 153] [75, 90, 116, 122, 142, 154] [64, 91, 105, 123, 143, 155] [81, 92, 122, 124, 144, 156] [66, 93, 103, 125, 145, 157]

Code ID 159-37-5 · download JSON · raw on GitHub